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Space Missions Enabling SETI Searches
Farther and Farther Out

Dr. Claudio Maccone

SETI Committee and Interstellar Space Exploration Committee,
International Academy of Astronautics (IAA)

Address: Via Martorelli, 43, I-10155 Torino (TO) - Italy
E-mail: cmaccone @ to.alespazio.it

ABSTRACT

As for an ET presence within the Solar System, this author believes that no such presence was scientifically proven so far.

As for future SETI Project, he thinks that, currently, three basic ways of doing SETI from space can be envisaged:

1. Using large (> 10 meters) space antennas orbiting the Earth, like the Japanese VSOP satellite already launched February 12, 1997 for doing VLBI in conjunction with other antennas on Earth. These large antennas could be advantageously used for rejecting the Radio Frequency Interference (RFI) that is the main problem currently plaguing the SETI searches made from the ground.

2. Doing SETI from the far side of the Moon, where the Moon body acts like a shield preventing man-made RFI from the Earth interfere with the expected incoming ETI signal. Since 1993, an international group of scientists (inspired by the work of Dr. Jean Heidmann) published a series of good papers on this project, called "Crater Saha Project" from the name of the far side crater selected to do SETI searches.

3. To use gravitational lensing effect created by the Sun as a "focussing device" to strengthen very weak ETI signals that our present day apparatus would not be able to detect. This implies sending a spacecraft endowed with a large inflatable antenna (50 to 100 meters) outside the solar system, at least at a distance of 550 Astronomical Units (AU). This is about 14 times the distance of Pluto from the Sun, or, alternatively, 3.17 light days. A formal Proposal to let the FOCAL space mission become a reality was already submitted by this author to the European Space Agency (ESA) in Paris in 1993. Now this mission is being studied also by the NASA experts at the Jet Propulsion Laboratory.

A MORE EXTENDED DESCRIPTION

ANSWERING QUESTION 1A.   Since 1960 (Project Ozma, led by Frank D. Drake), quite a few SETI searches have been conducted by the most distinguished SETI professional radio astronomers based in the most technologically advanced countries. Still, in agreement with all scientific published material on these SETI searches, this author believes that no ETI signal could be detected with absolute reliability about what one would expect to come with it: its precise location on the celestial sphere and its distance, a central frequency (like the hydrogen line at 1420 MHz) around which Doppler shifts would occur, and an elaborate texture including the ETI "message". It is claimed in this prize essay that this "lack of actual contact up to now" is probably because we could not reach far enough with our small earth-based radio telescopes. In fact, only those stars located within a sphere of about 100 light years radius from the Sun have been searched for ETI so far. This 100 light years range is a ridiculously small distance when compared to the size of our own Galaxy, a hundred thousand light years across in diameter. It's like Christopher Columbus hadn't discovered America just because he couldn’t go further west than the Canary Island! So, the limited range of today's SETI searches is a bottleneck that must be overcome somehow. But how can one overcome it? Actually, 100 light years away is just about as far as we can reach with our ground-based radio telescopes simply because their collecting area cannot be increased any further: that is, larger and heavier antennas would collapse under their own weight. So, what about forgetting about ground-based antennas and going over to space instead?

ANSWERING QUESTION 1B.  It is claimed in this prize essay that, currently, three basic ways of doing SETI from space can be envisaged:

1. Using large space antennas orbiting the Earth, higher than, say 10 meters in diameter, like the Japanese VSOP satellite already launched February 12, 1997 for doing VLBI (Very Large Baseline Interferometry) in conjunction with other antennas on Earth. These large antennas could be advantageously used for rejecting the Radio Frequency Interference (RFI) that is the main problem currently plaguing the SETI searches made from the ground.

2. Doing SETI from the far side of the Moon, where the advantage is that the body of the Moon acts like a shield preventing man-made RFI from the Earth interfere with the expected incoming ETI signal. Since 1993, an international group of scientists (inspired by the work of Dr. Jean Heidmann of the Paris Meudon Observatory) published a series of good papers on this project, called "Crater Saha Project" from the name of the Saha crater on the far side of the moon that was selected to conduct such SETI searches.

3. To use gravitational lensing effect created by the mass of the Sun as a "focussing device" to strengthen very weak ETI signals that our present day apparatus would not be able to detect. This implies sending a spacecraft endowed with a large inflatable antenna (50 to 100 meters in diameter) quite far outside the solar system, actually at least at a distance of 550 Astronomical Units (AU). This is about 14 times the distance of Pluto from the Sun, or, alternatively, 3.17 light days. A formal Proposal to let the FOCAL space mission become a reality was already submitted by this author to the European Space Agency (ESA) in Paris in 1993. But now this mission is being taken seriously also by the NASA experts at the Jet Propulsion Laboratory.

Finally, there is one more "mathematical suggestion" that the author would like to put forward in this prize essay: this is the replacement of the good old Fast Fourier Transform (FFT) by another Transform called the Karhunen-Loève Transform (KLT) that seems to be better adapted to SETI searches than the FFT. In fact, the KLT adapts itself to any background noise distribution (whereas the FFT does so only for white noise) and works for wide band signals as well as for narrow band signals.

A SHORT INTRODUCTION TO DOING SETI FROM EARTH ORBIT

Doing SETI from the Earth orbit would essentially solve one problem: the rejection of Radio Frequency Interference (RFI) that is currently plaguing all SETI searches made from the ground.

As a matter of fact, even the most recent and technologically advanced SETI searches, like Project Phoenix (since 1994 led by Dr. Jill C. Tarter of the SETI Institute in Mountain View, CA), always had to face the RFI rejection problem as the most challenging problem in today's experimental SETI. It is true that brilliant solutions to RFI rejection for SETI were indeed found, like the use of two widely separated antennas (one at Parkes and one at Mopra) in the 1995 — 6 Project Phoenix searches made from Australia. However, the enormous increase telecommunications of all kinds on Earth is clearly increasing the man-made RFI. So, all SETI searches currently made from the ground are doomed to become unfeasible sooner of later, say in the time of a few decades.

Going over to orbits around the Earth may be a solution to the RFI problem. As a matter of fact, the first large antennas to orbit the Earth were initially designed in the 1980's and 1990's by dedicated teams working at both American and European space companies, with the supervision of NASA and ESA (the European Space Agency) experts. For instance, the QUASAT (an acronym for QUAsar SATellite) was a joint NASA-ESA Project intended to put a 12-meter antenna into a low Earth orbit to do VLBI (Very Large Baseline Interferometry) from both space and the ground at the same time. QUASAT was to have four observing radio frequencies: 327 MHz, 1.66 GHz (the frequency of the OH radical maser), 5 GHz and 22 GHz (the frequency of the water maser). As you see, neutral the hydrogen line at 1420 MHz (21 cm wavelength) was not included as an observing frequency, and so QUASAT would not have been a "good" SETI searches satellite. Still the technology developed for QUASAT was going to be very much the same as the technology requested for all future SETI satellites, all essentially made up by a large antenna. The person writing these lines was a member of the Team of Aeritalia (later renamed Alenia Spazio) that designed QUASAT for NASA and ESA. Thus, he knows that QUASAT was to be an inflatable antenna of the type that he would have later suggested for the FOCAL spacecraft, described hereafter in this Essay. However, QUASAT never could become a reality because of budget cuts made by NASA in 1988, and so the project was abandoned. To Soviets, wishing to do better than the Americans and European, then designed their own version of QUASAT, that they named RADIOASTRON. But this also became an ill-fated Project when the Soviet Union collapsed in 1991. So, it was finally the Japanese who made it.

Successfully launched February 12, 1997, the Japanese spacecraft, initially named VSOP and later renamed HALCA, is the first example in history of space-tested large antenna, 8 meters in diameter. It works beautifully for VLBI purposes in conjunctions with many radio telescopes on the ground all over the world. But SETI is not part of the HALCA science goals at all: the "only" purpose of HALCA is to produce largely magnified radio pictures of Quasars and Radio Galaxies by virtue of the increased baseline, which enables one to achieve a resolution much higher than obviously achieved with Earth-based antennas alone. HALCA just has problems in working with the 22 GHz water maser line because the accuracy of the antenna surface is not high enough, and this is so because the HALCA's paraboloid is a tensional structure held together by thin ropes tied to expandable booms.

That's not all about Space VLBI, though. Wishing to do better than the Japanese, the Americans are now designing ARISE, a newer and larger space antenna about 25 meters in diameter. It would observe even on frequencies much higher than the water maser at 22 GHz, such as 43 GHz and even 81 GHz. However, sad to say, even ARISE is not going to be a satellite for doing SETI from space.

Why is that so? Why is SETI from space so neglected?

Well, the answer is that SETI altogether is often neglected by the science community all over the world. The problem about SETI is not just a question of technical details about how to detect the ETI signals. The problem is actually a problem of culture. SETI searches are still confused by the non-scientifically trained people as UFO-hunting. And since the official SETI scientists have always denied any connection between SETI and UFO's, then the vast majority of would-be-supporters for SETI has somehow been uncertain about getting a more serious interest into SETI itself. The more so since the NASA SETI Program, that had been ongoing since 1979, was drastically terminated by Congress in October 1993.

So, SETI as of 1998 is a serious scientific endeavor that has hardly enough money to support itself, both from the various national Science Institutions and Space Agencies all over the world, and from the lay people.

It is true, however, that "the learned international élite" is doing something to keep the dream alive.

For instance, the International Academy of Astronautics (IAA), based in Paris, has a SETI Committee to which belong about fifty of the best SETI experts from all over the world. This SETI Committee, currently chaired by Dr. Jill C. Tarter of the SETI Institute, organizes every year in October two SETI Sessions that take place as a part of the International Astronautical Federation (IAF) Congress. These two Sessions are named, respectively:

1. SETI Science and Technology;

2. SETI Interdisciplinary Connections.

As their names say, they are respectively devoted to the technicalities of SETI and to the larger sociological, philosophical, religious and political issues that go along with how Humankind would be struck by the announcement that "we are not alone".

A SHORT INTRODUCTION TO DOING SETI FROM THE FAR SIDE OF THE MOON

Doing SETI from the far side of the Moon would solve once and forever the problem of RFI rejection that is increasingly plaguing all current SETI searches made from the ground. In fact, the far side of the Moon is obviously shielded by the body of the Moon itself from all radio transmissions and noises produced by Humans on Earth.

More correctly still, the far side of the Moon is the only such shielded place near to Earth, and so it acquires an even higher political and moral interest: it seems the duty of politicians to defend this "radio oasis" from wild exploitation driven by purely commercial interest. The International Institute of Space Law (IISL), together with the International Academy of Astronautics (IAA) and the International Astronautical Federation (their Fellow Members count the best thousand space scientists and engineers from all over the world), has recently taken formal steps to "defend" the far side of the Moon against wild commercial exploitation. So is also starting doing COSPAR, the Committee on Space Research that is equally made up by over a thousand space-interested scientists.

But what should be better done in the practice?

The leading French SETI scientist, Jean Heidmann of the Paris Observatory in Meudon, gave the first practical hint in 1993: he selected just one crater on the far side of the Moon, named Crater Saha after the discoverer of the Saha equation used in plasma physics, as the best possible location where to set up a radio telescope on the far side. Crater Saha is located just a few degrees (3 to 5) south of the Moon equator, and it is just a little beyond the Moon line diving the near side from the far side. This location is important inasmuch as one could conceive of laying a transmission line (made by optical fibers) on the surface of the Moon between crater Saha and the nearby Mare Smithii, that is a plain looking toward the Earth. This "surface link" would be instrumental in order to let us get on Earth all the radio signals that the SETI antenna in Crater Saha would collect for us, free of any RFI. This "cable" made by optical fibers would actually have to be about 350 kilometers long because of the "Moon librations" (this is an apparent movement of the Moon as seen from the Earth, actually due to the fact that the orbit of the Moon around the Earth is not circular but, rather, slightly elliptical, so that an apparent "rotation" of and angle of 7.5 degrees appears to take place at every revolution of the Moon around the Earth).

But… setting up a 350 km optical fiber cable on the surface of the Moon would be very expensive if done by astronauts: something like a new Apollo mission would be required for this purpose and so there could be little hope that any space agency like NASA or ESA would ever fund any space mission of that type. Some new idea from today's astronautics is in order.

Thus, this author tried to solve the problem by planning an unmanned space mission based upon tether technology and thus called SETIMOON TETHER. Essentially, the idea is to put into orbit around the Moon and in the Moon equatorial plane a tethered system made up by three parts:

1. The folded up antenna intended to land inside Crater Saha. This would be shaped like a pile a folded phase arrays (i. e. planar " receiving surfaces" that, by virtue of a computer would do the same as a large parabolic dish would).

2. The folded, smaller antenna to land onto Mare Smythii and intended to keep the radio link with the Earth.

3. The "tether" in between the two end-points made up by the two above folded antennas. This would actually act as a tense, mechanical rope between the two end-points during the descent phase on the Moon surface, and would later become the "optical fiber link" between the antenna in Saha and the one on Mare Smithii, so that no astronaut's work would be required to set up the system.

The soft landing of the two end-points on their respective targets would be insured by use of airbags of the type that worked so well during the Mars Pathfinder landing of July 4, 1997: in fact, the gravity on the Moon is just about a half the gravity on Mars, and so the soft landing by virtue of airbags should work quite well. Finally, a slightly modified landing strategy would call for the landing of the whole set of two folded antennas plus the tether onto Mare Smythii first. Then, from there, a small, carried-along missile would launch the folded array into Crater Saha. But again the folded array would be tied up by a tether to the antenna on Mare Smythii, and this tether would then land automatically on the surface and become the set of optical fibers connecting the two end-points.

Apart from all technical details, the setting up of a SETI facility inside Crater Saha will most likely create political issues like:

1. Can we protect the radio environment of, say, one third of the far Moon side around Crater Saha by forbidding all RFI production by law?

2. If the answer to the above question is "yes", then this would mean that only satellites in polar orbit around the Moon not in sight of Saha would be tolerated: all other orbits would have to be forbidden, because the apparatus on board the satellites would produce its own noise, spoiling the far side of the Moon in the half-plane tangent to the Moon in Saha and above it.

3. Who is going to take these political decisions? Probably the United Nations, but we have to pave them the way for this.

AN INTRODUCTION TO THE "FOCAL" SPACE MISSION CONCEPT

Two foci of the gravitational lens of the Sun are predicted to exist by the general theory of relativity (see Figure 1):

1. A focus for electromagnetic waves, located along a line starting at a distance of 550 Astronomical Units (AU), that is 3.17 light days, or 14 times the distance from the Sun to Pluto. It shall be proved that any point beyond this minimal distance is a focus also. Thus, any spacecraft that can fly to 550 AU and beyond can take full advantage of the huge radio magnifications of any astronomical object lying on the other side of the Sun with respect to the spacecraft position.

2. A focus for gravitational waves and neutrinos located within the Solar System at distances 22.45 and 29.59 AU (roughly between the distances of the orbits of Uranus and Neptune). The physical justification for the existence of this focus is that:

1. a gravitational wave can penetrate through the Sun because such waves scatter significantly only in the presence of significant mass density rather than the charge on electrons which scatter electromagnetic waves, and
2. the bulk of the Sun's mass is more highly concentrated within its inner layers than within its    outer layers, i.e. the Sun's radial density is maximum at the center and zero at the surface.

Figure 1:Comparison between the paths of electromagnetic waves and gravitational waves being focussed by the gravitational field of the Sun.

The Proposed FOCAL Space Mission to 550 AU and Beyond to 1000 AU

In the sequel of this prize essay I shall deal with the FOCAL space mission, intended to let a spacecraft travel to 550 AU and beyond, up to 1000 AU, to enable Humankind to see the terrific, magnified radio pictures of whatever lies on the other side of the Sun with respect to the spacecraft position.

A time-optimized astrodynamical design of the spacecraft trajectory to reach a distance of 550 A.U. is clearly vital for the success of this mission, inasmuch the faster the spacecraft the shorter the mission. Such a time-optimized mission design, however, is no simple task, primarily because it must be achieved in conjunction with a specially designed propulsion system. Three propulsion systems are currently being investigated for the purpose of very deep space missions: conventional chemical engines, nuclear-electric propulsion and solar sailing. Each has its own advantages, and the question of making a final choice for one of them is basically an engineering question that will not be discussed in this prize essay.

Thus putting aside the question of selecting the propulsion system for the FOCAL mission, one is still left to face the following purely astrodynamical problem: discover the sequence of flybys of all planets and of the Sun enabling the spacecraft to exit the solar system at the highest possible speed and along a given direction, i.e. pointing toward a given point of the celestial sphere. I shall call this the "maximum terminal velocity problem".

To my knowledge, the above problem has never been solved or even just thoroughly discussed in mathematical terms, though it was described for the first time by the German-American astrodynamicist Krafft von Ehricke in 1972 (refs. [1], [2]). In Chapter 3 an optimized Jupiter flyby is described that could be used to get a useful preliminary mission design.

Notice also that a Sun flyby is vital to let the spacecraft exit the Solar System at any high inclination to the ecliptic.

Finally, recent studies of solar sails, particularly by Giovanni Vulpetti prove that sails used in a Sun flyby at about the minimal ‘safe’ distance of 0.30 AU (roughly the orbit of Mercury) could best achieve this goal at a small mission cost without the constraints of any planetary launch window.

These are clearly topics of much vital interest for the coming years, and will have to explored in much more depth than it was possible to do in this prize essay.

Missions to Uranus and Neptune to Detect Focussed Gravitational Waves

I now shall deal with two space missions to detect the Sun's focused gravitational waves by sending spacecraft to distances of 22.45 and 29.59 AU (ref. [3]). At first sight, these missions may not seem of interest to SETI, but deeper reflection show that this is not the case because neutrinos also could be used by ET to send signals around.

First of all, these missions are obviously much less time-demanding than reaching 550 AU. Moreover, the Galactic Center is probably the richest sources of gravitational waves in the Galaxy.

The first space mission I call WAVE-URANUS. Its scientific justification lies in that, about the year 2030, Uranus will be positioned, on the celestial sphere, 7 to 8 degrees south of the line of sight of the star Elnath in the constellation Auriga, which is almost on the line of sight of the Galactic Anticenter. Thus, Humankind will experience the unique opportunity of having Uranus almost exactly aligned with the Sun and with the Galactic Center. Humankind is still in time not to miss this opportunity for a great space mission. In fact, a heliocentric Hohmann transfer to Uranus takes about 16.16 years. Thus, launching the spacecraft in 2012 would allow it to reach Uranus in 2028. And, nearby Uranus, it could study the flow of gravitational waves from the Galactic Center that are focussed about that location.

I also propose a similar WAVE-NEPTUNE mission. In fact, Neptune also will be almost aligned with the Sun and the Galactic Center, but that will occur only about 2061 A.D.. Thus, a launch in 2030, followed by the requested heliocentric Hohmann transfer to Neptune (lasting 30.78 years), would bring the spacecraft at the right position to further pursue the investigations on gravitational waves and neutrinos already started by the previous mission to Uranus.

REFERENCES

1. K. A. Von Ehricke, "Saturn-Jupiter Rebound - A method of High-Speed Spacecraft Ejection from the Solar System", Journal of the British Interplanetary Society, 25 (1972), pp. 561-571.
2. G. Matloff and K. Parks, "Interstellar Gravity Assist Propulsion: A Correction and A New Application", Journal of the British Interplanetary Society, 41 (1988), pp. 519-526.
3. D. Sonnabend, "To the Solar Foci", Ph. D. Thesis published by Caltech, JPL Publication 79-18, Pasadena, California, June 1, 1979.

 

Chapter 1

SO MUCH GAIN AT 550 AU

1.1 Introduction

The gravitational focussing effect of the Sun is one of the most amazing discoveries produced by the general theory of relativity. The first paper in this field was published by Albert Einstein in 1936 (ref. [1]), but his work was virtually forgotten until 1964, when Sydney Liebes of Stanford University (ref. [2]) gave the mathematical theory of gravitational focussing by a galaxy located between the Earth and a very distant cosmological object, such as a quasar.

In 1978 the first "twin quasar" image, caused by the gravitational field of an intermediate galaxy, was spotted by the British astronomer Dennis Walsh and his colleagues. Subsequent discoveries of several more examples of gravitational lenses eliminated all doubts about gravitational focussing predicted by general relativity.

Von Eshleman of Stanford University then went on to apply the theory to the case of the Sun in 1979 (ref. [3]). His paper for the first time suggested the possibility of sending a spacecraft to 550 AU from the Sun to exploit the enormous magnifications provided by the gravitational lens of the Sun, particularly at microwave frequencies, such as the hydrogen line at 1420 MHz (21 cm wavelength). This is the frequency that all SETI radio astronomers regard as "magic" for interstellar communications, and thus the tremendous potential of the gravitational lens of the Sun for getting in touch with alien civilizations became obvious.

The first experimental SETI radio astronomer in history, Frank Drake (Project Ozma, 1960), presented a paper on the advantages of using the gravitational lens of the Sun for SETI at the Second International Bioastronomy Conference held in Hungary in 1987 (ref. [4]), as did Nathan "Chip" Cohen of Boston University (ref. [5]). Non-technical descriptions of the topic were also given by them in their popular books (refs. [6] and [7]).

However, the possibility of planning and funding a space mission to 550 AU to exploit the gravitational lens of the Sun immediately proved a difficult task. Space scientists and engineers first turned their attention to this goal at the June 18, 1992, Conference on Space Missions and Astrodynamics organized in Turin, Italy, lead by me. The relevant Proceedings were published in 1994 in the Journal of the British Interplanetary Society (ref. [8]). Meanwhile, on May 20, 1993 I also submitted a formal Proposal to the European Space Agency (ESA) to fund the space mission design (ref. [9]). The optimal direction of space to launch the FOCAL spacecraft was also discussed by Jean Heidmann of Paris Meudon Observatory and myself (ref. [10]), but it seemed clear that a demanding space mission like this one should not be devoted entirely to SETI. Things like the computation of the parallaxes of many distant stars in the Galaxy, the detection of gravitational waves by virtue of the very long baseline between the spacecraft and the Earth, plus a host of other experiments would complement the SETI utilization of this space mission to 550 AU and beyond. The mission was dubbed "SETISAIL" in earlier papers (ref. [11]), and "FOCAL" in the proposal submitted to ESA in 1993.

1.2 The Minimal Focal Distance of 550 AU for Electromagnetic Waves

The well-known Schwarzschild solution to the Einstein field equations is the mathematical foundation upon which the theory of the gravitational lens of the Sun rests. From it a long string of formulae can be developed. Since those formulae are derived in standard textbooks, I shall simply rewrite without proofs the basic equations I need to explain the advantages provided by the gravitational lens of the Sun, suggesting the interested reader consult the basic refs. [8], [12] and [13] for the relevant mathematical demonstrations.

The geometry of the Sun gravitational lens is easily described: incoming electromagnetic waves (arriving, for instance, from the center of the Galaxy) pass outside the Sun and pass within a certain distance r of its center. Then the basic result following from the Schwarzschild solution shows that the corresponding deflection angle a (r) at the distance r from the Sun center is given by

. (1.2-1)

Figure 1.2 depicts the various parameters.

Figure 1.2: Basic geometry of the gravitational lens of the Sun, showing the minimal focal length and the FOCAL spacecraft position.

 The light rays, i.e. electromagnetic waves, cannot pass through the Sun's interior (whereas gravitational waves and neutrinos can), so the largest deflection angle a occurs for those rays just grazing the Sun surface, i.e. for . This yields the inequality

(1.2-2)

with

. (1.2-3) 

From the illustration it should be clear that the minimal focal distance is related to the tangent of the maximum deflection angle by the formula

. (1.2-4)

Moreover, since the angle is very small (its actual value is about 1.75 arc seconds), the above expression may be rewritten by replacing the tangent by the small angle itself:

. (1.2-5)

Eliminating the angle between equations (1.2-3) and (1.2-5), and then solving for the minimal focal distance , one gets

. (1.2-6)

This basic result may also be rewritten in terms the Schwarzschild radius

, (1.2-7)

yielding

. (1.2-8)

Numerically, one finds

. (1.2-9)

This is the fundamental formula yielding the minimal focal distance of the gravitational lens of the Sun, i.e. the minimal distance from the Sun's center that the FOCAL spacecraft must reach in order to get magnified radio pictures of whatever lies on the other side of the Sun with respect to the spacecraft position.

Furthermore, a simple, but very important consequence of the above discussion is that all points on the straight line beyond this minimal focal distance are foci too, because the light rays passing by the Sun further than the minimum distance have smaller deflection angles and thus come together at an even greater distance from the Sun.

And the very important astronautical consequence of this fact for the FOCAL mission is that it is not necessary to stop the spacecraft at 550 AU. It can go on to almost any distance beyond and focus as well or better. In fact, the further it goes beyond 550 AU the less distorted the collected radio waves by the Sun Corona fluctuations. The important problem of Corona fluctuations and related distortions is currently being studied by Von Eshleman and colleagues at Stanford University.

I would like to add here one more result that is very important because it holds well not just for the Sun, but for all stars in general. This I'll do without demonstration; that can be found on page 55 of ref. [12]. Consider a spherical star with radius and mass , which will be called the "focussing star". Suppose also that a light source (i.e. another star or an advanced extraterrestrial civilization) is located at the distance from it. Then ask how far is the minimal focal distance on the opposite side of the source with respect to the focussing star center? The answer is given by the formula

. (1.2-10)

This is the key to gravitational focussing for a pair of stars, and may well be the key to SETI in finding extraterrestrial civilizations. It could also be considered for the magnification of a certain source by any star that is perfectly aligned with that source and the Earth: the latter would then be in the same situation as the FOCAL spacecraft except, of course, it is located much further out than 550 AU with respect to the focussing, intermediate star. Finally, notice that equation (1.2-10) reduces to equation (1.2-6) in the limit , i.e. (1.2-6) is the special case of (1.2-10) for light rays approaching the focussing star from an infinite distance.

1.3 The (Antenna) Gain of the Gravitational Lens of the Sun

Having thus determined the minimal distance of 550 AU that the FOCAL spacecraft must reach, one now wonders what's the good of going so far out of the solar system, i.e. how much focussing of light rays is caused by the gravitational field of the Sun. The answer to such a question is provided by the technical notion of "antenna gain", that stems out of antenna theory.

A standard formula in antenna theory (sometimes named after J. C. Slater) relates the antenna gain, , to the antenna effective area, , and to the wavelength l or the frequency n

by virtue of the equation (refer, for instance, to ref. [13], in particular page 6-117, equation (6-241)):

>. (1.3-1)

Now, assume the antenna is circular with radius , and also assume 50% efficiency. Then, the antenna effective area is obviously given by

. (1.3-2)

Substituting this back into (1.3-1) yields the antenna gain as a function of the antenna radius and of the observed frequency

. (1.3-3)

The important point here is that the antenna gain increases with the square of the frequency, thus favoring observations on frequencies as high as possible.

Is anything similar happening for the Sun's gravitational lens also? Yes is the answer, and the "gain" (one maintains this terminology for convenience) of the gravitational lens of the Sun can be proved to be

(1.3-4)

or, invoking the expression (1.2-7) of the Schwarzschild radius

. (1.3-5)

The mathematical proof of equation (1.3-4) is difficult to achieve.

The author, unsatisfied with the treatment of this key topic given in refs. [1], [3] and [13], turned to three engineers of the engineering school in his home town, Renato Orta, Patrizia Savi and Riccardo Tascone. To his surprise, in a few weeks they provided a full proof of not just the Sun gain formula (1.3-4), but also of the focal distance for rays originated from a source at finite distance, equation (1.2-10). Their proof is fully described in ref. [12], and is based on the aperture method used to study the propagation of electromagnetic waves, rather than on ray optics.

Using the words of these three authors’ own Abstract, they have "computed the radiation pattern of the [spacecraft] Antenna+Sun system, which has an extremely high directivity. It has been observed that the focal region of the lens for an incoming plane wave is a half line parallel to the propagation direction starting at a point [550 AU] whose position is related to the blocking effect of the Sun disk (Figure 1.2). Moreover, a characteristic of this thin lens is that its gain, defined as the magnification factor of the antenna gain, is constant along this half line. In particular, for a wavelength of 21 cm, this lens gain reaches the value of 57.5 dB. Also a measure of the transversal extent of the focal region has been obtained. The performance of this radiation system has been determined by adopting a thin lens model which introduces a phase factor depending on the logarithm of the impact parameter of the incident rays. Then the antenna is considered to be in transmission mode and the radiated field is computed by asymptotic evaluation of the radiation integral in the Fresnel approximation".

1.4 The Combined, Total Gain upon the FOCAL Spacecraft

One is now able to compute the Total Gain of the Antenna+Sun system, which is simply obtained by multiplying equations (1.3-3) and (1.3-5)

(1.4-1)

Since the total gain increases with the cube of the observed frequency, it favors electromagnetic radiation in the microwave region of the spectrum. The table in Figure 1.4 shows numerical data provided by equations (1.3-5) and (1.3-3) for five selected frequencies: the hydrogen line at 1420 MHz and the four frequencies that the Quasat radio astronomy satellite planned to observe, had it been built jointly by ESA and NASA as planned before 1988 (ref. [14]) (the definition of dB is: ).

Figure 1.4: Table showing the gain of the Sun’s lens alone, the gain of a 12-meter spacecraft (S/C) antenna and the combined gain of the Sun+S/C Antenna system the at five, selected frequencies important in radio astronomy.

1.5 The Image Size at the Spacecraft Distance z

The next important notion to understand is the size of the image of an infinitely distant object created by the Sun lens at the current spacecraft distance z from the Sun (). We may define such an image size as the distance from the focal axis (i.e. from the spacecraft straight trajectory) at which the gain is down 6 dB. The formula for this (proven in ref. [8]) is

(1.5-1)

Thus the image size increases with the spacecraft distance z from the Sun. The table in Figure 1.5 provides a quantitative feeling of how the image size changes with the spacecraft distance from the Sun.

Figure 1.5: Table showing image sizes for a 12 meter antenna, located at distances of 550AU, 800 AU and 1000 AU from the Sun, for the five selected frequencies.

It is clear that these image size values are very small compared to the spacecraft distance from the Earth. This means that if we want to observe a certain point source in the sky, the alignment between this source, the Sun and the spacecraft position must be extremely precise. In fact, the spacecraft tracking must exceed by far what we are able to do within the solar system today. However, this is not true if the source we want to observe is the center of the Galaxy, which is a very broad source: slight changes in the spacecraft trajectory (say in a spreading spiral shape) would enable us to gradually see much of the galactic center at the huge resolution provided by the gravitational lens of the Sun.

1.6 Requirements on the Image Size and Antenna Beamwidth at the Spacecraft Distance z

There are two "geometrical" requirements that must be fulfilled in order that the combined lens system Sun+FOCAL spacecraft antenna can work at best:

1) Size Requirement: the full antenna dish of the FOCAL spacecraft must fall well inside the cylindrical region centered along the focal axis and having radius equal to r_6dB. That is, the spacecraft feed-dish radius must be considerably smaller than r_6dB

(1.6-1)

2) Angle Requirement: the impact-radius circle around the Sun within which electromagnetic waves are focussed towards the FOCAL spacecraft must fall well within antenna beamwidth of the FOCAL spacecraft. In a little more technical terms, the Half-Power Beam Width (=HPBW, i.e. the angular width of the main lobe of the spacecraft antenna at the half-power level) should be considerably greater than the angle subtended at the spacecraft distance by twice the incident ray impact radius at the Sun

(1.6-2)

The Tables in Figures 1.6.1 and 1.6.2 show that both these conditions are fulfilled at the three FOCAL distances from the Sun for the five selected frequencies, respectively.

Figure 1.6.1: Table showing image sizes vs. the antenna radius for a 12 meter antenna located at various distances from the Sun for the five selected frequencies.

Figure 1.6.2: Table showing HPBW vs. aspect angle of the Sun for a 12-meter antenna located at various distances from the Sun for the five selected frequencies.

1.7 The Angular Resolution at the Spacecraft Distance z

The notion of angular resolution of the Sun lens is relevant to the discussion. Angular resolution is simply defined as the ratio of the image size (at the spacecraft distance z from the Sun) to that distance z. From eq. (1.5-1),

(1.7-1)

Clearly the angular resolution also depends on the spacecraft distance z from the Sun, and it actually improves (i.e. it gets smaller) as long as the distance increases beyond 550 AU.

Figure 1.7: Table showing angular resolution for three spacecraft distances (550 AU, 800 AU and 1000 AU), at the five selected frequencies.

The table in Figure 1.7 gives angular resolutions for the same three distances at the same five frequencies. Let us take a moment to ponder over these numbers. The best angular resolutions achieved so far, in visible light, were obtained by the European astrometric satellite Hipparcos, launched in 1989, and dismissed from service in 1993. Though the apogee kick motor of Hipparcos didn't fire, forcing technicians to take the software originally written for a circular geostationary orbit and re-write it for a highly elliptical orbit, the Hipparcos mission has proven a success. The resolutions achieved by Hipparcos are at a level of 2 milliseconds of arc precision. Checking this figure against the above table, one can see that the gravitational lens of the Sun plus a (modest) 12-meters antenna would improve the angular resolution by about three orders of magnitude (at radio frequencies).

1.8 The Spatial Resolution at the Spacecraft Distance z

Finally, let us turn to the spatial resolution, simply called the resolution, of an astronomical object we want examine by help of the gravitational lens of the Sun. It defined by

. (1.8-1)

Again, beyond 550 AU the resolution improves (i.e. the angle gets smaller) slowly with the increasing spacecraft distance from the Sun. The table in Figure 1.7 shows the spatial resolutions for a very wide range of distances, from the Oort Cloud to cosmological objects like quasars.

Figure 1.8: Table showing the spatial resolutions for astronomical objects at selected distances from the Sun, for a 12-meter spacecraft antenna.

REFERENCES

[1] A. Einstein, "Lens-like Action of a Star by the Deviation of Light in the Gravitational Field", Science, Vol. 84, (1936), pages 506-507.

[2] S. Liebes, Jr., "Gravitational Lenses", Physical Review, Vol. 133 (1964), pages B835-B844.

[3] V. Eshleman, "Gravitational Lens of the Sun: Its Potential forObservations and Communications over Interstellar Distances", Science, Vol. 205 (1979), pages 1133-1135.

[4] F. Drake, "Stars as Gravitational Lenses", Proceedings of the Bioastronomy International Conference held in Balatonfüred, Hungary, June 22-27, 1987, G. Marx editor, pp. 391-394.

[5] N. Cohen, "The Pro's and Con's of Gravitational Lenses in CETI", Proceedings of the Bioastronomy International Conference held in Balatonfüred, Hungary, June 22-27, 1987, G. Marx editor, p. 395.

[6] F. Drake and D. Sobel, Is Anyone Out There?, Delacorte Press, New York, 1992, see in particular pp. 230-234.

[7] N. Cohen, Gravity's Lens, Wiley Science Editions, New York, 1988.

[8] C. Maccone, "Space Missions Outside the Solar System to Exploit the Gravitational Lens of the Sun", in Proceedings of the International Conference on Space Missions and Astrodynamics held in Turin, Italy, June 18, 1992, C. Maccone editor, Journal of the British Interplanetary Society, Vol. 47 (1994), pages 45-52.

[9] C. Maccone, "FOCAL, A New space Mission to 550 AU to Exploit the Gravitational Lens of the Sun", A Proposal for an M3 Space Mission submitted to the European Space Agency (ESA) on May 20, 1993, on behalf of an international Team of scientists and engineers. Later (October 1993) re-considered by ESA within the "Horizon 2000 Plus" space missions plan.

[10] J. Heidmann and C. Maccone, "AstroSail and FOCAL: two extraSolar System missions to the Sun's gravitational focuses", Acta Astronautica, Vol. 35 (1994), pp. 409-410.

[11] C. Maccone, "The SETISAIL Project", in ‘Progress in the Search for Extraterrestrial Life", Proceedings of the 1993 Bioastronomy Symposium held at the University of California at Santa Cruz, 16-20 August 1993, G. Seth Shostak editor, Astronomical Society of the Pacific Conference Series, Volume 74 (1995), pages 407-417.

[12] R. Orta, P. Savi and R Tascone, "Analysis of Gravitational Lens Antennas", in Proceedings of the International Conference on Space Missions and Astrodynamics held in Turin, Italy, June 18, 1992, C. Maccone editor, Journal of the British Interplanetary Society, Vol. 47 (1994), pages 53-56.

[13] John D. Kraus, Radio Astronomy, 2nd edition, Cygnus-Quasar Books, Powell, Ohio, 1966, in particular pages 6-115 through 6-118.

[14] A. Hawkyard and A. Anselmi, "QUASAT Industrial Phase A Study", Executive Summary, Aeritalia GSS Report QS-RP-A1-0004 (1988).

 

Chapter 2

SETI AND THE "FOCAL" SPACE MISSION

2.1 Introduction

SETI, or the Search for Extraterrestrial Intelligence, started in 1959 with the seminal paper of Cocconi and Morrison (ref.[1]). SETI was experimentally pursued for the first time by Drake in 1960 (Project Ozma, ref. [2], Reading #25) and later developed into a large body of interdisciplinary knowledge (refs. [3], [4], [5], [6], [7]).

The central problem of experimental SETI is to recover weak radio signals out of noise. In isolation, the problem would not be difficult to solve with the aid of modern filtering algorithms and computers, but there are a number of complications:

1) We don't know the radio frequency at which the extraterrestrials may be trying to communicate with us.

2) We don't know from which direction in the sky the signals may be reaching us.

3) We don't know how to distinguish a "natural" signal, i.e. a radio emission caused by some astrophysical mechanism, from an "intelligent" signal, i.e. a radio emission intentionally broadcast by ETs whose civilization has achieved a level of technological development comparable or more advanced than ours.

4) In case we do detect a non-natural radio signal, it is not clear how we will deduce its meaning.

Tentative solutions to the above four complications listed above have been provided by the world-wide community of experimental SETI-radio astronomers consisting mainly of Americans, but also Russians (formerly Soviets), French, Dutch, Australians, Argentinians, Japanese, Italians, etc.) over the years since 1960. In summary,

1) As to the frequencies to examine, all frequencies between about 1 and 10 GHz are suited for interstellar communications, but those in the range 1-2 GHz seem to be the most appropriate ones. In particular, "magic" frequencies (i.e. optimal frequencies for communication because all "ET radio astronomers" must know their numerical values) are supposed to be the neutral hydrogen line (1420 MHz) and the hydroxyl lines (1612, 1615, 1667 and 1720 MHz). The portion of spectrum between these lines is nicknamed the waterhole, and galactic communications would "meet" around the waterhole as animals gather around water ponds in draught.

2) This problem of which direction to look in was being solved at NASA by means of the All Sky Survey, started at the Goldstone 70-meter antenna on October 12, 1992 but terminated abruptly by the U.S. Congress in October 1993. A previous all-sky search had been started by the Planetary Society in 1985 by resorting to more modest antennas located in Massachusetts and Argentina: this was META Project, later technically upgraded over the years up to the currently ongoing BETA II Project. Apart from these two projects, the vast majority of SETI searches were targeted on stars expected to harbor life because they are similar to the Sun.

3) To distinguish a "natural" signal from an "intelligent" signal we look at the signal intensity profile around the signal's central frequency. If the profile is Gaussian, the signal is expected to be natural, if the profile is a very narrow peak (almost comparable to a Dirac delta function) then the signal is expected to have been broadcasted by a technologically advanced civilization. This we like to call the narrowband assumption in SETI and I shall discuss it in the next Section.

4) To understand the meaning of an alien intelligent signal we probably need some Cosmic Language based on a mathematical scheme integrated with knowledge from other branches of the sciences, such as physics, chemistry and biology. Hans Freudenthal’s Lincos (Design of a Language for Cosmic Intercourse) was the first human attempt in this direction (1960).

2.2 The Narrow-Band Assumption in SETI

All SETI searches carried on thus far have been for narrow-band signals, simply because SETI radio astronomers believe that an extraterrestrial civilization wishing to make itself know all over the Galaxy would broadcast radio signals easily distinguishable from natural emissions. Since natural emissions have a Gaussian intensity profile around their own central frequency, ET would replace this Gaussian by a Dirac delta function, i.e. an (almost) infinitely narrow peak, to let us understand that it was an "intelligent" being, rather than Mother Nature, to send us that wave package. We shall call this wide-spread belief the narrow-band assumption in SETI.

No mathematical proof of the narrow-band assumption seems to have been given. I would like to provide one here, based on the well-known information theory put forward by Claude Shannon in 1948. For this proof, I am going to extend an argument given in 1964 by the leading Russian SETI expert Nikolai S. Kardashev. His paper (ref. [8]) was seminal, at least in that it put forward the classification of extraterrestrial civilizations as Type I, II and III, according to whether the extraterrestrials were able to funnel the energy of their own planet, solar system, or galaxy, respectively. In that paper Kardashev also used Shannon's formula for the rate of information transmission within a certain information channel over the frequency band between and :

(2.2-1)

where and are the power spectral densities of the useful signal and noise, respectively. Kardashev states that "by solving the appropriate variational problem, we may show the maximum rate of information transmission to be achieved under the condition

. (2.2-2)

Here and are the bounds of the transmitter's transmission band. It is accordingly quite clear that the spectrum of the artificial source must show the reversed-shaped parabola of the equation

. (2.2-3)

i.e. the spectrum of the artificial radio emission must feature a maximum. Kardashev omitted the mathematical steps leading from (2.2-1) to (2.2-3) because they are mathematically trivial. Yet I would like to show that expressing them explicitly pays off, inasmuch as it offers a mathematical proof of the narrow-band assumption universally adopted by SETI searchers. I do so by taking the variation of (2.2-1) with respect to the unknown function together with the two normalization conditions fulfilled by the signal and noise, namely,

and (2.2-4)

where and are the total signal and noise power over the given bandwidth, respectively. This results in the variational equation

(2.2-5)

where and are Lagrange multipliers. Performing the differentiation under the integral sign, one has

(2.2-6)

which has a solution of the form

. (2.2-7)

Next the Lagrange multiplier must be determined by integrating both sides of the solution (2.2-7) with respect to between and , and then invoking the normalization conditions (2.2-4)

(2.2-8)

The result is

. (2.2-9)

Substituting this into (2.2-7) gives

. (2.2-10)

This formula can also be given in different form by recalling the definition of the extraterrestrial transmission bandwidth:

and . (2.2-11)

Equation (2.2-10) then yields

and (2.2-12)

whence Kardashev's formula (2.2-3) (or equation (4) in ref. [8], Reading #28) is obtained at once.

Now we want to use (2.2-10) to prove the narrow-band assumption. The argument is as follows: an extraterrestrial civilization wishing to make itself known would try to send as much information as possible about itself. It would thus try to maximize the information transmission rate (2.2-1) over the transmission bandwidth of their apparatuses. Then (2.2-10) shows that, keeping both the total signal and the noise powers ( and ) fixed over the given bandwidth, the narrower this bandwidth is, the cleared the signal spectral density stands out against the noise spectral density , i.e.

. (2.2-13)

In conclusion, ETs must transmit over the narrowest possible bandwidths to let their messages be understood clearly against the background noise. And on Earth one must use very narrowband spectral analyzers to detect ET candidate signals. Over the years the bandwidths that humans are using have steadily decreased to 1 Hz and even less: hundreds or thousands of Hz are now achievable by dedicated computers like those of Project Phoenix (formerly NASA-SETI Project) and the BETA 2 system run at Harvard by the Planetary Society.

2.3 A Simple Introduction to the KLT

Understanding the mathematical model of a physical fact may be difficult to people who are not familiar with the required mathematical background. Yet mathematics is just the correct language by which physics and engineering achieve success. Translating this mathematical language into the language of "common" words may be desirable whenever a mathematical advance is made that has to be described to newcomers in "easy terms".

This section is devoted to a rather new mathematical tool that may improve our understanding of physical phenomena: the Karhunen-Loève Transform, hereafter abbreviated KLT. Essentially, it is something superior than the classical Fourier Transform (FT). To explain why, let me use a comparison in classical mechanics. Consider an object, for instance a book, and a three-axis rectangular reference frame, oriented arbitrarily with respect to the book. Now all mechanical properties of the book itself are described by a 3x3 (symmetric) matrix called "inertia matrix" (or "inertia tensor") whose elements are, in general, non-zero. Handling a matrix whose elements are all non-zero is obviously more complicated than handling a matrix where all elements are zeros except for those lying on the main diagonal (this is called a "diagonal matrix"). Thus one may be led to wonder whether a certain axes transformation exists that changes the inertia matrix of the book into a diagonal matrix.

Classical mechanics shows that only one special orientation of the rectangular frame with respect to the book exists, yielding a diagonal inertia matrix: the three axes must coincide with a set of three vectors (parallel to the book edges) called "eigenvectors" or "proper vectors" of the book. In other words, each body possesses an intrinsic set of three rectangular axes, called "eigenvectors" of the body, that describes its mechanical properties most simply. This is referred to as "diagonalizing the matrix."

Now let me go to signal processing, which is our interest here. By adding random noise to a deterministic signal one obtains what is called a "noisy signal" or, in case the power of the signal is much smaller than the power of the noise "a signal buried into the noise". Since the noise+signal is a random function of the time, one can describe it by a statistical quantity called autocorrelation (or simply "correlation"), defined as the mean value of the product of the values of X(t) at two different instants and , that is . This correlation, obviously symmetric in and , may play just the same role as the inertia matrix in the book example. Thus, if one seeks for the eigenvectors of the correlation, and then changes the reference frame to the new set of vectors, the easiest possible description of the signal+noise is achieved. This is the key idea behind the KLT.

On may also look at the KLT from a slightly different point of view. In mathematical physics the well-known "method of normal coordinates" allows one describe the "small oscillations" of a dynamical system in the best possible way by expressing the Lagrangian as a sum of Lagrangians, each of them representing a simple harmonic oscillator. This is the result of a "principal axes" transformation, i.e. a Lagrangian coordinate transformation that yields the separation of variables naturally. The KLT is just the statistical version of that.

2.4 Mathematics of the KLT

The KLT (ref. [9]) is named after two mathematicians, the Finn, K. Karhunen and the French-American, M. Loève, who proved independently and roughly simultaneously (1946) that the series (2.4-1) hereafter is convergent. Put it this way, the KLT looks like a purely mathematical topic, but this is not, of course, the case. Using the language of engineers and radio astronomers, we say that it is possible to represent the signal+noise X(t) as the infinite series (K-L, or KLT, expansion)

(2.4-1)

Assuming that the noise (auto)correlation is a known function of and it can be proved that the functions are the eigenfunctions of the correlation, namely the solutions to the integral equation

. (2.4-2)

These form an orthonormal basis in the Hilbert space, and they actually are the optimal basis to describe the noisy signal, better than any classical Fourier basis. One can thus say that the KLT adapts itself to the shape of the signal+noise, whatever it is.

A further advantage of KLT is that the in (2.4-1) are orthogonal random variables, i.e. that . If X(t) is a Gaussian process, this orthogonality amounts to statistical independence, meaning that the terms in the KLT expansion are uncorrelated. Since the constants are both the (all positive) eigenvalues and the variances of the random variables , any KLT expansion, when truncated to keep only the first few terms, is the best approximation to the original function in the mean square sense.

Finally, the mathematical theory of KLT shows that the process need not be stationary. This too spells the difference against the classical Fourier techniques, that hold rigorously true for stationary processes only.

2.5 KLT for SETI

The narrowband assumption was the rationale behind all ETI radio searches made thus far all over the world. Consequently, only Fourier Transform (FT) or Fast Fourier Transform (FFT) techniques were used to find the very narrow bandwith (called "bin" in the SETI jargon) in which an unusual amount of received radio energy might indicate the presence of a signal, either sinusoidal or pulsed.

In this section, however, we would like to maintain that the traditional usage of the FFT within SETI might sooner or later be replaced by the adoption of the KLT. This is no new idea. In 1983 the French SETI radio astronomer, François Biraud, was the first person within the SETI community to describe the advantages of KLT over FFT for detecting wideband signals, ref. [10]. Apart from the technical issue, there seem to another and deeper one: adopting the KLT means to be ready for the unexpected. Indeed, we know nothing about the nature of ETI signals: we have just made a set of "reasonable" assumptions, and we are trying to see whether they serve to put us in touch with the rest of the universe. Enlarging the possible signals to look for by shifting from FFT to KLT can only help.

Very promising work on the KLT for SETI was done by Robert S. Dixon and Charles A. Klein, both with the Ohio State University in Columbus, Ohio, ref. [11]. After acknowledging that the KLT is more general than the FT because it makes no assumption about the signal periodicity or waveform, these authors took one important new step in pointing out that only the largest of the KLT components need be calculated, in contrast to the FT, where all components (one for each frequency) must be calculated. This largest KLT component, or coefficient, is what the mathematicians call the "dominant eigenvalue" in the solution of the integral equation (2.4-2). Dixon and Klein did not attempt to prove any mathematical theorem about this fact, but they did numerical computer experiments showing that this must be the case.

One might ask what prevents radio astronomers from using KLT for SETI now. The simple answer is the computational burden. In fact, the KLT kernel is the correlation, and, being the mean value of the product of two random variables, this kernel is not separable. In general, one cannot hope for the existence of a fast KLT algorithm. In turn, this means that the computer time required to calculate the eigenvalues and eigenvectors of a correlation matrix of order N is proportional to , rather than to as for FFT.

Nevertheless, several concurrent developments seem to be paving new ways to overcome the above difficulties. On the one hand, the steady improvements in computer hardware and parallelization techniques seem to lead to very fast algorithms capable of getting the eigenvalues and eigenvectors of large square symmetric matrix as the correlation. On the other hand, the progress in the mathematical theory of the KLT has progressed steadily since the early 1950s, and I mention some results of potential interest for SETI applications:

1) For an exponential correlation of the form

(2.5-1)

the problem of finding the KLT was completely solved as back as 1958, ref.[12], pages 99-101.

2) The correlation (2.5-1) is just an example of stationary random process, i.e. a process having a correlation of the form

. (2.5-2)

Now for the general stationary correlation (2.5-2), Srinivasan and Sukavanam (refs. [13],[14]) obtained a solution to the integral equation (2.4-2), where the right-hand side is an arbitrary real-valued function f(...) defined on the positive real axis. They assumed that f(...) admits a Laplace Transform , and in the practice confined themselves to the case where the latter is given by

(2.5-3)

though they state that their arguments can be easily extended to the more general case where admits a Mittag-Leffler expansion. For the case of (2.5-3), they gave explicit, though complicated, formulas for computing the eigenvalues numerically, but apparently not for computing the eigenfunctions. This prevents further study to be carried on about the possibility that a fast K-L algorithm might exist for the stationary correlation (2.5-2). More investigations are needed.

3) S. Watanabe, ref. [15], introduced the method of the K-L expansion into the realm of pattern recognition in 1965. In this application, an image is to be represented in terms of an optimal coordinate system, and the set of basis vectors which make up this coordinate system is referred to as an eigenpicture. The basis vectors are simply the eigenfunctions of the covariance matrix of the ensemble of images. The state-of-the-art in the application of the KLT to images is described in ref. [16], and this shows that computers already exist that are powerful enough to apply the KLT to image processing.

4) The first fast K-L algorithm (ref. [17]) was obtained in 1976 by A. K. Jain. An excellent description of his mathematical algorithm was given by A. Rosenfeld and A. C. Kak in ref. [18]. The key idea is to make the correlation separable by resorting to exponential functions. For instance, let an image belonging to a given set of images (random field) be sampled on a square sampling lattice, and let denote the samples, where both m and n take on integer values from through . Then the assumed correlation is of the type

(2.5-4)

where , , and are constants, the former two being less than unity. For this (discrete) correlation both the K-L eigenvalues and eigenfunctions may be explicitly found, as in ref. [17]. There Jain has shown that if the image boundary pixels are known, they may be used to modify the rest of the image in such a way as to possess a K-L transform that can be implemented using FFT (or the more recently developed fast sine transform). Thus Jain's result is essentially a reduction of KLT to FFT preserving the typical advantages of both.

2.6 Conclusion: Advantages of the KLT for the FOCAL Space Mission

Since the 1950s the number of applied scientists using the KLT for their research has slowly increased, but the KLT still lies outside the realm of most current scientific research. This situation of neglect seems to have been caused primarily by two obstacles:

1) The exceedingly heavy computational burden required by the KLT;

2) The many obscure points still plaguing the KLT mathematical theory.

While the first obstacle might be overcome relatively soon by the development of parallel processing computers, paving the mathematical way requires more effort.

SETI is the field of research where the KLT might distinguish itself in comparison to the FFT. Only the KLT, in fact, would reveal wide band signals whatever the nature of the noise spectrum, and whether or not the random process is stationary. The time appears to be ripe for the KLT to be taken seriously by the SETI as well as by other signal-processing investigators all over the world.

There is, however, an additional and very important point that I would like to stress: the KLT is not just used for filtering weak signals out of the noise: the KLT is used for data compression also.

In fact, consider the four basic steps of the KLT:

1) Find the eigenvectors of the autocorrelation of the set of data.

2) Assume the set of eigenvectors as new vectors.

3) Expand the set of data over this set of vectors and then truncate it by (arbitrarily) declaring that whatever is beyond a certain (low) correlation value is "noise", i.e. unnecessary data.

4) Reverse transform to the original set of axes and reconstruct the "filtered" set of data.

The data compression occurs at step 3).

Moreover, since data compression is essential in the radio link at huge distances like those to be reached by FOCAL, one concludes that the KLT is the bestpossible way of compressing the data sent by FOCAL to the Earth, i.e. the best way of letting FOCAL be success.

REFERENCES

1. G. Cocconi and P. Morrison, "Searching for Interstellar Communications", Nature, Vol. 184 (1959), page 844.
2. D. Goldsmith, The Quest for Extraterrestrial Life - A Book of Readings, University Science Books, Mill Valley, California, 1980.
3. "The Search for Extraterrestrial Life: Recent Developments", Proceedings of the International Astronomical Union Symposium No. 112, held in Boston, June 18-21, 1984, editor Michael D. Papagiannis, D. Reidel Publishing Company, Dordrecht, The Netherlands, 1985.
4. "Bioastronomy - The Next Steps", Proceedings of the 99th Colloquium of the International Astronomical Union, held in Balatonfüred, Hungary, June 22-27, 1987, editor George Marx, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988.
5. "Bioastronomy - The Search for Extraterrestrial Life - The Exploration Broadens", Proceedings of the Third International Symposium on Bioastronomy held at Val Cenis, Savoie, France, 18-23 June 1990, editors Jean Heidmann and Mike Klein, Lecture Notes in Physics No. 390, Springer Verlag, Berlin-Heidelberg, 1991.
6. "Progress in the Search for Extraterrestrial Life", Proceedings of the Fourth International Symposium on Bioastronomy held at the University of California at Santa Cruz, 16-20 August 1993, G. Seth Shostak editor, Astronomical Society of the Pacific Conference Series, Volume 74 (1995).
7. "Astronomical and Biochemical Origins and the Search for Life in the Universe", Proceedings of the Fifth International Conference on Bioastronomy (IAU Colloquium No.161) held at Capri, Italy, 1-5 July 1996, C. Cosmovici, S. Bowyer and D. Werthimer editors, Editrice Compositori, Bologna, Italy, 1997.
8. N. S. Kardashev, "Transmission of Information by Extraterrestrial Civilizations" reading #28 in ref. [2].
9. C. Maccone, Telecommunications, KLT and Relativity, IPI Press, Colorado Springs, 1994, ISBN # 1-880930-04-8.
10. F. Biraud, "SETI at the Nançay Radio telescope", Acta Astronautica, 10, 759-760 (1983).
11. Robert S. Dixon and Charles A. Klein, "On the Detection of Unknown Signals", paper presented at the USA-USSR Joint Conference on the Search for Extraterrestrial Intelligence, held at the University of California at Santa Cruz, August 5-9, 1991.
12. Wilbur B. Davenport and William R. Root, An Introduction to the Theory of Random Signals and Noise, McGraw-Hill, New York, 1958.
13. S. K. Srinivasan and S. Sukavanam, "Photo-count statistics of Gaussian light of arbitrary spectral profile", J. Phys. A-5, 682-694 (1972).
14. S. K. Srinivasan, Stochastic Point Processes and Their Applications, Griffin, London, 1974, in particular pages 83-87 and 95-98.
15. S. Watanabe, "Karhunen-Loève expansion and factor analysis theoretical remarks and applications", Proc. 4th Prague Conf. Inform. Theory, 1965.
16. M. Kirby and L. Sirovich, "Application of the Karhunen-Loève Procedure for the Characterization of Human Faces", IEEE Trans. on Pattern Analysis and Machine Intelligence, 12, 103-108 (1990).
17. A. K. Jain, "A fast Karhunen-Loève transform for a class of random processes", IEEE Trans. Commun. COM-24, 1023-1029 (1976).

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